What is the general solution for (3cos(x)+2sin(x))^2+5sin^2(x)=9-3sqrt(3) ?

The general solution for (3cos(x)+2sin(x))^2+5sin^2(x)=9-3sqrt(3) is x=(2pi)/3+pin,x=(5pi)/6+pin

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(3cos(x)+2sin(x))^2+5sin^2(x)=9-3sqrt(3)

(3 cos (x) + 2 sin (x))^{2} + 5 sin^{2} (x) = 9 - 33

x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

Solution steps

(3 cos (x) + 2 sin (x))^{2} + 5 sin^{2} (x) = 9 - 33

Subtract

9 - 33

from both sides

9 cos^{2} (x) + 12 cos (x) sin (x) + 9 sin^{2} (x) - 9 + 33 = 0

Rewrite using trig identities

33 + 12 \cdot \frac{sin ( 2 x )}{2} = 0

12 \cdot \frac{sin ( 2 x )}{2} = 6 sin (2 x)

33 + 6 sin (2 x) = 0

Move

33

to the right side

6 sin (2 x) = - 33

Divide both sides by

6

sin (2 x) = - \frac{3}{2}

General solutions for

sin (2 x) = - \frac{3}{2}

2 x = \frac{4 π}{3} + 2 πn, 2 x = \frac{5 π}{3} + 2 πn

Solve

2 x = \frac{4 π}{3} + 2 πn : x = \frac{2 π}{3} + πn

Solve

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{6} + πn

x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

7 cos (2 x) = 7 sin^{2} (x) + 5

2 cos^{2} (x) - sin (x) = - 1

\frac{cos ( x ) \cdot cot ( x )}{1 - sin ( x )} = 3

\frac{sin ( 8 )}{30} = \frac{sin ( θ )}{120}

114 = sin (x)

What is the general solution for (3cos(x)+2sin(x))^2+5sin^2(x)=9-3sqrt(3) ?
The general solution for (3cos(x)+2sin(x))^2+5sin^2(x)=9-3sqrt(3) is x=(2pi)/3+pin,x=(5pi)/6+pin